Theorem elre0re 39230

Description: Specialized version of 0red 10637 without using ax-1cn 10588 and ax-cnre 10603. (Contributed by Steven Nguyen, 28-Jan-2023.)

Assertion

Ref	Expression
elre0re	⊢ (𝐴 ∈ ℝ → 0 ∈ ℝ)

Proof of Theorem elre0re

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ax-rnegex 10601	. 2 ⊢ (𝐴 ∈ ℝ → ∃𝑥 ∈ ℝ (𝐴 + 𝑥) = 0)
2		readdcl 10613	. . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝑥 ∈ ℝ) → (𝐴 + 𝑥) ∈ ℝ)
3		eleq1 2899	. . . 4 ⊢ ((𝐴 + 𝑥) = 0 → ((𝐴 + 𝑥) ∈ ℝ ↔ 0 ∈ ℝ))
4	2, 3	syl5ibcom 247	. . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝑥 ∈ ℝ) → ((𝐴 + 𝑥) = 0 → 0 ∈ ℝ))
5	4	rexlimdva 3283	. 2 ⊢ (𝐴 ∈ ℝ → (∃𝑥 ∈ ℝ (𝐴 + 𝑥) = 0 → 0 ∈ ℝ))
6	1, 5	mpd 15	1 ⊢ (𝐴 ∈ ℝ → 0 ∈ ℝ)

Syntax hints:   → wi 4   ∧ wa 398   = wceq 1536   ∈ wcel 2113  ∃wrex 3138  (class class class)co 7149  ℝcr 10529  0cc0 10530   + caddc 10533

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-ext 2792  ax-addrcl 10591  ax-rnegex 10601

This theorem depends on definitions:  df-bi 209  df-an 399  df-ex 1780  df-cleq 2813  df-clel 2892  df-ral 3142  df-rex 3143

This theorem is referenced by:  rernegcl  39277  renegadd  39278  reneg0addid2  39280  resubeulem1  39281  resubeulem2  39282  resubeu  39283  remul02  39311  remul01  39313  readdid1  39315  resubid1  39316  renegneg  39317  relt0neg1  39320  relt0neg2  39321